On the effect of misclassification on bias of perfectly measured covariates in regression

This note clarifies under what conditions a naive analysis using a misclassified predictor will induce bias for the regression coefficients of other perfectly measured predictors in the model. An apparent discrepancy between some previous results and a result for measurement error of a continuous variable in linear regression is resolved. We show that similar to the linear setting, misclassification (even when not related to the other predictors) induces bias in the coefficients of the perfectly measured predictors, unless the misclassified variable and the perfectly measured predictors are independent. Conditional and asymptotic biases are discussed in the case of linear regression, and explored numerically for an example relating birth weight to the weight and smoking status of the mother.